A218858 Number of Gaussian primes at taxicab distance n from the origin.

0, 0, 4, 12, 0, 16, 0, 20, 0, 16, 0, 28, 0, 24, 0, 32, 0, 32, 0, 36, 0, 24, 0, 36, 0, 64, 0, 32, 0, 48, 0, 44, 0, 32, 0, 72, 0, 64, 0, 48, 0, 72, 0, 60, 0, 56, 0, 60, 0, 40, 0, 56, 0, 72, 0, 112, 0, 64, 0, 76, 0, 88, 0, 56, 0, 136, 0, 92, 0, 80, 0, 76, 0, 88, 0

Offset

0,3

Comments

Except for n = 2, there are no Gaussian primes at an even taxicab distance from the origin. All terms are multiples of 4. See A218859 for this sequence divided by 4.

The arithmetic derivative of Gaussian primes is either 1, -1, I, or -I.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

T. D. Noe, Linear plot

Example

In the taxicab distance, the four Gaussian primes closest to the origin are 1+I, -1+I, -i-I, and 1-I. The 12 at taxicab distance 3 are the four reflections of 3, 2+I, and 1+2I.

Mathematica

Table[cnt = 0; Do[If[PrimeQ[n - i + I*i, GaussianIntegers -> True], cnt++], {i, 0, n}]; Do[If[PrimeQ[i - n + I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 0, -1}]; Do[If[PrimeQ[i - n - I*i, GaussianIntegers -> True], cnt++], {i, 1, n}]; Do[If[PrimeQ[n - i - I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 1, -1}]; cnt, {n, 0, 100}]

Crossrefs

Cf. A055025 (norms of Gaussian primes).

Cf. A222593 (first-quadrant Gaussian primes).

Cf. A225071, A225072 (number of terms at an odd distance from the origin).

Sequence in context: A222316 A255383 A354777 * A014458 A099733 A350259

Adjacent sequences: A218855 A218856 A218857 * A218859 A218860 A218861

Keyword

nonn

Author

T. D. Noe, Nov 12 2012

Status

approved